Characterization for stable equivalence

If is Unital, then we can characterize the stable equivalence condition nicely. if and only if for some positive integer
Proof:
If for some , then To prove this, just apply the properties given above for the sum of two elements in the semigroup of projections.

The standard picture of , described in the two propositions below, is a concrete and useful description of the -group of a unital -algebra. Proposition 3.1.8 shows that proposition 3.1.7 form a Universal property of .